"De la/of the" implies ownership, as in "el perro de la chica"="the girl's dog". While DL's literal translation "the best player of the league" is awkward, "the league's best player" is natural English and accurate.

Some more cheats:
el ama de casa (the housewife)
el asma (asthma)
el arca (the ark)
el hambre (hunger)
el hampa (the underworld)
el arpa (the harp)
el águila (the eagle)
http://spanish.about.com/od/adjectives/a/el_for_la.htm
Your point about "el" when referring to people is a good one though. Escpecially when several occupations such as "artista", "dentista", or "modelo" do not change endings for gender, so gender is instead defined by the article.

I think this works in response to a question like "Who took that half court shot at the buzzer?" You could say "It was John", or "It was the the best player of the league" if you're not into the whole brevity thing.